World-sheet Scattering in AdS5xS5

Konstantin Zarembo

(Uppsala U.)

Integrability in Gauge and String Theory, Paris, 21.06.07

T.Klose, T.McLoughlin, R.Roiban, K.Z., hep-th/0611169

T.Klose, K.Z., hep-th/0701240

T.Klose, T.McLoughlin, J.Minahan, K.Z., 0704.3891

AdS/CFT correspondence Maldacena’97

Gubser,Klebanov,Polyakov’98

Witten’98

Spectral problem in large-N SYM:• can be reformulated in terms of an integrable spin

chain…

• … with unknown Hamiltonian (beyond O(λ))• but with exactly known S-matrix!

Minahan,Z.’02

Beisert,Kristjansen,Staudacher’03

Beisert,Staudacher’03

Beisert’05

Beisert,Eden,Staudacher’06

Beisert,Hernandez,Lopez’06

Sigma-model in AdS5xS5:• well-defined theory with known (although

complicated) Lagrangian

• also integrabl

• How does the (discrete) spin chain arise from the (continuous) world-sheet?

Bena,Polchinski,Roiban’03

Goal: compute the S-matrix on the world-sheet

explicitly to few first orders in perturbation theory

φ

yi’

t

zi

S5 AdS5

The metric

AdS5S5

Transverse coordinates:

+ RR flux

Sigma-model

Metsaev,Tseytlin’98

Sigma-model coupling constant:

Classical limit

is

Green-Schwarz-type coset PSU(2,2|4)/SO(4,1)xSO(5)

Fermion part of the sigma-model is

of Green-Schwarz type.

Conformal gauge is problematic:• no kinetic term for fermions • no holomorphic factorization for currents, …

RR flux requires manifest space-time supersymmetry

Standard CFT does not work…

Kinetic term for fermions:

If X=const, ρa=0 and the kinetic term vanishes.

Need to fix the light-cone gauge: X+=τ(then ρ0=Γ+ and the kin. term is not degenerate)

Gauge fixing

Light-like geodesics:

angular momentum

Berenstein,Maldacen,Nastase’02

Parnachev,Ryzhov’02

Callan,Lee,McLoughlin,Schwarz,Swanson,Wu’03

Arutyunov,Frolov,Plefka,Zamaklar’06

Unbroken bosonic subgroup:

Taking into account supersymmetry:

This also gets central extension with

p – world-sheet momentum

Beisert’05

Arutyunov,Frolov,Plefka,Zamaklar’06

World-sheet fields

World-sheet fields (embedding coordinates in AdS5xS5):

global time in AdS5 and angle on S5 – eliminated by gauge fixing

Fields form (2|2)x(2|2) multiplet of PSU(2|2)2:

Step 1: T-duality

- gauge parameterInterpolates between

temporal gauge:

and light-cone gauge:

How to fix the gauge?

Arutyunov,Frolov,Zamaklar’06

T-dual background

Metric:

B-field:

Step2: Nambu-Goto Lagrangian:

Step3: Gauge condition

Step 4: Rescaling

Gauge-fixed Lagrangian (the bosonic part):

Gauge-fixed action

Loop counting parameter:

Internal string length:

Decompatification (infinite string) limit:

Static gauge-fixed T-dual Nambu-Goto string:

• potenially problematic non-renormalizable field theory

• however in flat space perturbative calculcations in Nambu-Goto formalism (Lüscher,Symanzik,Weisz’80; Dietz,Filk’83;

Caselle,Hasenbusch,Panero’05) agree, upon ζ-function regularization, with conformal-gauge results (Olesen’85)

(e.g. at two loops Roiban,Tirziu,Tseytlin’07)

Bosonic Lagrangian in AdS5xS5 (up to quartic order in fields):

• Massive, integrable 2d field theory

• Lorentz-invariant kinetic terms• Lorentz invariance is broken by interactions• Gauge-dependent: a=1/2 (pure l.c. gauge) is the

simplest case

Spectrum

(8|8) massive states:

Loop corrections should give:

exact Beisert,Dippel,Staudacher’04;Beisert’05 two-loop correction!

World-sheet scattering

S-matrix:

Integrability (consistency with Yang-Baxter equations) requires:

SU(2)4 – invariace:

Beisert’05

up to an overall phase

Symmetries of the S-matrix

Bosonic (su(4)4) generators:

Supersymmetries:

Spin chain explanation: the chain is dynamical, its length may decrease on increase in the process of scattering

Beisert’05

Hopf algebra

Co-product:

promotes symmetries of the S-matrix

to non-trivial Hopf algebra:Gomez,Hernandez’06; Plefka,Spill,Torrielli’06

graded permutation

String explanation 1: non-locality of supercharges

Non-locality of supercurrents:local operator

Arutyunov,Frolov,Plefka,Zamaklar’06

By contour manipulations:

non-trivial co-product

Klose,McLoughlin,Roiban,Z.’06

non-local

String explanation 2: fractional statistics

If the world-sheet scattering states become solitons (giant magnons)

Hofman,Maldacena’06

Fermion number:

Conjecture: giant magnon with momentum

carries fractional fermion number

In principle, can be checked by summing over known fermion

modes Minahan’06; Papathanasiou,Spradlin’07 of the giant magnon Jackiw,Rebbi’76; Goldstone,Wilczek’81; Niemi,Semenoff’86

(Fermi statistics)

Bernard,Leclair’90; Fendley,Intriligator’91

and then:

Tree level

Tree-level factorization:

Scattering of bosons at tree level

Ym(p)Yn(p’) → Yk(p)Yl(p’)

SO(4) → SU(2)xSU(2)

Factorization works!

No terms, allowed by group theory

Explicit calculation yields:

Dressing phase

To leading order in :

Beisert,Hernandez,Lopez’06

Beisert,Eden,Staudacher’06

agrees with tree-level

scattering amplitudes in the sigma-model

Dorey,Hofman,Maldacena’07

Near-flat limitMaldacena,Swanson’06

perturbative string modes

giant magnons Hofman,Maldacena’06

“Left-moving” sector:

Reduced Sigma Model

Consistent truncation of the full sigma-model:

Maldacena,Swanson’06

- chiral SO(8) spinor

Dispersion relation

Exact dispersion relation:

In the near-flat limit:

At λ→∞:

(exact)

Beisert,Dippel,Staudacher’04

Tadpole cancellation

+ = 0

= 0

Yi’ (S5) Zi (AdS5)

(fermions)

The same mechanism renders the model finite

to any order in perturbation theory

Mass renormalization

=

Mass-shell condition:

S-matrix

One loop:

Two loops:

+ mass renormalization

+ two-loop corrections to Z-factors

Tree-level amplitude:

One-loop correction:

Klose,Z.’07

Two loops:

Klose,McLoughlin,Minahan,Z.’07

Agrees!

“Lorentz-invariant” form of the S-matrix

All Lorentz non-invariance of amplitudes in the reduced model can be

hidden in the momentum-dependent redefinition of the coupling:

In the full σ-model/spin chain, LI is possibly q-deformed rather than violated

Gomez,Hernandez’06; Young’07

Conclusions

• At strong coupling, the SU(2|2)xSU(2|2) S-matrix describes the scattering of the string modes in the light-cone gauge

• Perturbative calculations of scattering amplitudes agree with the conjectured BHL/BES phase up to two loops at strong coupling (and to four loops at week coupling Bern,Czakon,Dixon,Kosower,Smirnov’06)

• Ultimately we want to quantize the σ-model with periodic bounary conditions, when S-matrix is not well defined.

• S-matrix → asymptotic Bethe ansatz• Exact Bethe ansatz?